Method for predicting flow and performance characteristics of a body using critical point location

ABSTRACT

A method is provided for calculating flow performance characteristics of a body immersed in a fluid under a set of fluid flow conditions. The method comprises providing a geometrical description of a surface of the body and determining the set of fluid flow conditions. The set of fluid flow conditions includes a combination selected from the combination set consisting of angle of attack and leading edge stagnation point location, angle of attack and flow separation point location, and leading edge stagnation point location and flow separation point location. The method further comprises calculating a velocity flow field for the body using a critical point potential flow methodology and calculating flow performance characteristics.

CROSS-REFERENCE TO RELATED APPLICATIONS

This application claims priority to U.S. Provisional Application No.61/078,943, filed Jul. 8, 2008, which is incorporated herein byreference in its entirety.

FIELD OF THE INVENTION

The invention relates generally to performance prediction and design ofstructures immersed in a fluid regime. More particularly, the inventionrelates to a method of modeling aerodynamic forces and moments thatmakes use of measured or predicted locations of flow bifurcation points.

BACKGROUND OF THE INVENTION

Current design programs use two main methods to achieve the desiredaerodynamic characteristics for the airfoil. In the direct method, aniterative procedure is used to successively modify the shape of anairfoil until the desired aerodynamic characteristics are achieved. Inthe inverse method, the desired aerodynamic characteristics arespecified and the airfoil shape that meets these objectives isdetermined.

The set of governing equations used to solve the problem is generally asimplified version of the full Navier-Stokes equations (see Kuethe, A.M. and Chow, C. Y.: Foundations of Aerodynamics, John Wiley & Sons(1998)), which require large computer power and computing time even forsimple cases. The Euler equation is obtained when the fluid viscosity isneglected, and the Laplace equation results when the flow is assumed tobe irrotational. In the interest of simplicity and in order to reducethe demands on computational resources and time, most airfoil design andanalysis methods use either the Euler or Laplace equations. The velocityand pressure fields are determined by solving these equations. Theaerodynamic forces and moments are then related to the pressuredistribution around the airfoil.

In order to solve the problem for given free-stream conditions, it isnecessary to prescribe the boundary conditions at the fluid-structureinterface on the airfoil surface. They are usually specified either interms of the flow direction tangential to the airfoil surface (the“Dirichlet condition”) or in terms of the normal derivatives of the flowat the airfoil surface (the “Neumann condition”). See Woods, L. C.: TheTheory of Subsonic Plane Flow, Cambridge University Press, (1961)(“Woods Reference”). A unique solution is usually obtained by imposingthe condition that (1) the flow leaves the trailing edge smoothly and(2) the velocity is zero or finite at the trailing edge. See Katz, J.and Plotkin, A.: Low-Speed Aerodynamics, Cambridge University Press,(2001). This condition is generally known as the “Kutta condition.” Theviscous effects using various boundary layer approximations and modelingof phenomena such as laminar-to-turbulent transition are then superposedon the potential flow solution in an iterative manner to determine theresultant velocity and pressure field around the airfoil. Theaerodynamic forces and moments generated by the airfoil are thenobtained by integrating the pressure around the airfoil.

The procedure is repeated to obtain the aerodynamic characteristics atother flow conditions.

A number of computer programs and associated software have beendeveloped using such methods. The differences between various programsmay be generally attributed to the relative quality of the viscousmodeling used to determine boundary layer characteristics. These designmethods have been used to design airfoils with predictable aerodynamiccharacteristics within a certain range of Reynolds numbers at low anglesof attack. However, errors in the computed performance are known toincrease significantly in the presence of flow separation near stall andpost-stall conditions. Further, the generally used methods fail topredict hysteresis when an airfoil undergoes pitching oscillations.

The most widely used methods for airfoil design and analysis start witha potential flow model and incorporate viscous effects in one of twoways: (1) Eppler's PROFILE (see Eppler, R.: Airfoil Design and Data,Springer-Verlag (Berlin), 1990); and (2) Drela's XFOIL airfoil analysismethod (see Drela, M., and Giles, M. B.: ISES: A Two-Dimensional ViscousAerodynamic Design and Analysis Code, AIAA Paper 87-0424, January 1987).In the original Eppler airfoil analysis method, the pressuredistribution is first obtained with a potential flow model. Thispressure distribution is used to compute viscous effects using boundarylayer approximations and superimposed on the potential flow solution toaccount for changes in pressure distribution due to viscous effects. Theaerodynamic coefficients are then computed from the resulting pressuredistribution. An alternate approach used in Drela's XFOIL code attemptsto interactively couple viscous effects with the potential flowsolution. Most other methods follow similar basic concepts.

The potential flow method is based on solving the governing Laplaceequation, which is a second-order linear partial differential equation.The flow direction (α) in the free stream in relation to a referenceaxis is assumed known. It requires two independent boundary conditionsto obtain, out of infinite possible combinations, one unique solution.

Two types of boundary conditions are generally considered. In theNeumann type conditions, zero-flow normal to a solid surface isspecified. In the Dirichlet type conditions, the flow about a body isuniquely determined if the flow direction θ is completely known on thegiven profile surface. The flow direction θ is measured from the chordline direction, a convenient reference line fixed in airfoil. If θ_(S)is the value of θ on the airfoil surface, then if the leading edgestagnation point (LESP) S1 and the flow separation point (FSP) S2 arefixed in position on the airfoil surface, θ_(S) is completely known, andthe flow direction at all points in the flow is fixed. In particular,θ_(∞), the flow direction at ∞, is fixed, as the incidence α is theangle between the airfoil chord (reference line) and the streamdirection at ∞, α=θ_(∞). Thus corresponding to each pair of locationsS1, S2 there is a unique value of α. The pressures over the airfoil arealso uniquely determined, and therefore the lift L is fixed in value.Conversely, if L and alpha are prescribed the positions S1 and S2 willbe uniquely determined. More generally we need to prescribe any pair ofthe quantities alpha, L, S1, S2 to obtain a unique Dirichlet flow (seeWoods Reference).

In both cases, the problem is usually resolved by assuming Kuttacondition at the trailing edge. The Kutta condition at the trailing edgesimultaneously provides the basis for calculating the circulationgenerated by the lifting body. Such an approach has been known and usedfor over a century and has provided an excellent first approximation forthe aerodynamic forces and moments generated by a lifting body (seeTheodorsen, T. and Garrick, I. E.: General Potential Theory of ArbitraryWing Sections. NACA TR 452 (1933)), when the lift coefficient variedalmost linearly with α. This approach gave poor results in the presenceof extensive flow separation on the suction side of the airfoil upstreamof the trailing edge at high α, near and beyond stall (see de Vargas, L.A. T., de Oliveira, P. H. I., de Freitas Pinto, R. L. U., Bortoulus, M.V., and e Souza, M. da Silva.: Comparison between modern procedures foraerodynamic calculation of subsonic airfoils for applications in lightaircraft designs, Proc. of COBEM, 18th International Congress ofMechanical Engineering, November 2005, Ouro Preto, MG.).

The current methods suffer from a fundamental mathematical drawback: theleading-edge stagnation point obtained initially by imposing the Kuttacondition at the trailing edge is not fully corrected in response toviscous corrections that are computed in an iterative manner. Since thepotential flow model is governed by elliptic equations (see Jeffrey, A.:Applied Partial Differential Equations. An Introduction. ElsevierScience, (2002)), where the downstream conditions have an impact on theupstream flow conditions, the presence of flow separation has a directimpact on the location of the leading-edge stagnation point and viceversa. This in turn has a significant impact on the aerodynamic forcesand moments generated by the airfoil. In the absence of extensive flowseparation (e.g., at low angles of attack), the error introduced byassuming a fixed leading-edge stagnation point location is marginal andpractically ignored both in experiments and computations. However, theerror becomes significant at high angles of attack, especially near andbeyond stall. As a result, these methods completely fail at and beyondstall when there is a real potential for generating high lift.

Thus the conventional potential flow model provides useful butrestricted solutions due to viscous effects that manifest themselves inthe boundary layer in the form of transition, turbulence, and flowseparation. On the other hand, full Navier-Stokes (N-S) equations couldbe used to solve the problem numerically but they require tremendouscomputational resources and time. Even then, a degree of empiricism isrequired to model and predict transition, turbulence, and flowseparation in solving N-S equations. Thus, strictly speaking, the N-Sapproach is semi-empirical in nature, though significantly more detailedand complex compared to the potential flow approach. In comparison, thepotential flow approach is simpler and, with proper physics and accuratemathematical modeling, it is capable of providing valuable engineeringsolutions with minimal computing resources and time.

SUMMARY OF THE INVENTION

The methods and systems of the invention provide improved predictionsfor the aerodynamic performance of wing sections, optimization of winddesigns based on aerodynamic performance requirements and constraints,and the capability to obtain aerodynamic performance estimates throughmeasurements of flow bifurcation points and freestream angle-of-attack.

An illustrative aspect of the invention provides a method of calculatingflow performance characteristics of a body immersed in a fluid under aset of fluid flow conditions. The method comprises providing ageometrical description of a surface of the body and determining the setof fluid flow conditions. The set of fluid flow conditions includes acombination selected from the combination set consisting of angle ofattack and leading edge stagnation point location, angle of attack andflow separation point location, and leading edge stagnation pointlocation and flow separation point location. The method furthercomprises calculating a velocity flow field for the body using acritical point potential flow methodology and calculating flowperformance characteristics. The calculated flow performancecharacteristics include at least one of the set consisting of angle ofattack, leading edge stagnation point location, flow separation pointlocation, lift coefficient, drag coefficient and pressure coefficientdistribution.

BRIEF DESCRIPTION OF THE DRAWINGS

The invention can be more fully understood by reading the followingdetailed description together with the accompanying drawings, in whichlike reference indicators are used to designate like elements, and inwhich:

FIG. 1 is a schematic representation of the flow around a wing section;

FIG. 2 is a flow diagram of a method for predicting flow performanceaccording to an embodiment of the invention;

FIG. 3 is a block diagram of a system for carrying out methods accordingto embodiments of the invention;

FIG. 4 is a flow diagram of a method for predicting flow performanceaccording to an embodiment of the invention;

FIG. 5 is a block diagram of a method for optimizing an airfoilaccording to an embodiment of the invention;

FIG. 6 is a block diagram of a control system according to an embodimentof the invention; and

FIG. 7 is a flow diagram of a method for predicting flow performanceaccording to an embodiment of the invention.

DETAILED DESCRIPTION OF THE INVENTION

While the invention will be described in connection with the preferredembodiment, it will be understood that it is not intended to limit theinvention to those embodiments. On the contrary, it is intended to coverall alternatives, modifications and equivalents that may be includedwithin the spirit and scope of the invention as described.

An embodiment of the invention provides a method for predicting theperformance of a wing section. Another embodiment provides a method fordesigning a wing section given performance requirements and constraints.Still another embodiment provides a method for measuring aerodynamicperformance using flow bifurcation point measurements in conjunctionwith other sensor measurements.

As used herein, the term “flow bifurcation point” (FBP) means a locationon a body surface where the flow attaches to or separates from the body.As shown in the idealized representation of FIG. 1, an airfoil 10 mayhave several FBPs including the leading edge stagnation point (LESP) 20,flow separation point (FSP) 30, and flow reattachment point 40.

When extensive flow separation region exists on the airfoil uppersurface at high angles of attack and the Kutta condition is notsatisfied at the trailing edge, it is necessary to formulate the problemdifferently by taking flow separation into account a priori. It can bemathematically shown that, according to the Dirichlet conditions forpotential flow past airfoil, specifying any given pair of three boundaryconditions (i.e., LESP, FSP, and angle of attack (a) at infinity)uniquely determines the third quantity (see Woods Reference). In such acase, the entire flow field is also simultaneously determined.

Thus, the LESP is uniquely determined for a given α and FSP. From themeasurement point of view, flow separation location can be predicted ifα and LESP are known. For steady flows, the lift and moment coefficientsas well the pressure and velocity distributions around the airfoil arecompletely determined if any one pair of the aforementioned threequantities are known (see Woods Reference).

Of course, the FSP may not be known a priori. What are generally knownare the direction of the free stream flow in relation to a referenceline and the geometry of the airfoil section whose chord is taken asthis reference line. It should be noted that for conventional approacheswhere the Kutta condition is applied and boundary layer effects aresuperimposed on the basic airfoil shape, this reference line loses itsoriginal meaning because the separated wake significantly alters theeffective shape of the airfoil with α, and the definition of the chordline becomes vague.

The methods of the present invention use more realistic physical andmathematical modeling. In an embodiment of the invention, themathematical model is based on potential flow theory combined withconformal transformation to describe the flow outside the separated flowregion on a body such as an airfoil or wing. There are several importantelements of this embodiment of the invention that will be discussed inthe context of an airfoil immersed in a fluid.

First, the Kutta condition is imposed only on the lower (pressure side)of the airfoil while flow separation is allowed to occur upstream of thetrailing edge on the upper surface. Second, assumptions based on freestreamline theory are used to describe the wake downstream of flowseparation from the upper and lower surfaces. Third, conformaltransformation is used to determine the flow field with prescribedsingularities at the leading edge, separation point, trailing edge andat infinity (wake). Fourth, the airfoil geometry is maintained as aninvariant. The superposition of boundary layer thickness does notrequire any change in airfoil geometry for potential flow computations.The mathematical approach only requires the location of the FSP in orderto determine the LESP for a given α. Finally, if both the LESP and FSPare specified (measured), α is automatically known. On the other hand,if α and LESP are specified, the FSP can be determined even withoutboundary layer calculations for a given airfoil geometry. Morecomplicated cases of flow separation and reattachment, e.g., separationbubble, can be derived using potential flow and conformal mappingtechniques providing similarly unique relationships for those cases.

FIG. 2 illustrates a generalized method M100 for determining the flowand performance characteristics for an airfoil immersed in a fluid undera set of flow conditions. The method M100 begins at S5 and at S10 asurface model of the airfoil is established. The airfoil model is amathematical representation of the airfoil surface. This model may beconstructed by connecting line segments between points within acoordinate system. The airfoil can also be described using equations;e.g., equations for thickness as a function of chord line distance abouta mean camber line. At S20, flow parameters are specified or otherwiseobtained. The flow parameter input includes information such asfreestream velocity, Reynolds numbers, kinematic viscosity, α, andrelated parameters. At S30, resolution parameters are provided orestablished. These parameters provide the spatial and numericalresolution for resolving the flow field and flow bifurcation points.

At S40, the input parameters and the surface model are used to calculatea velocity flow field. In the methods of the invention, this isaccomplished using the Critical Point Method (CPM), which is describedin more detail hereafter. At S50, the velocity field is used to obtainthe pressure field and airfoil performance characteristics (e.g., liftand moment), which may be displayed at S60. The method M100 ends at S65.

FIG. 3 depicts an airfoil performance processing system 200 configuredfor implementing the general method M100 according to an embodiment ofthis invention. The system 200 includes a performance data processor 210having an input receiving module 212, a critical point model computationmodule 214, and a flow field calculation module 216. There are threeinputs: airfoil surface model input 202, flow parameter input 204, andresolution input 206. As discussed above, the airfoil model input is amathematical representation of the surface that may be in the form ofcoordinate, connecting line segments between points, or mathematicalequations. The flow parameter input may include freestream velocity,Reynolds numbers, kinematic viscosity, .alpha., and related parameters.The resolution input establishes the spatial and numerical resolutionfor resolving the flow field and flow bifurcation points. The inputreceiving module 212 may be a user interface or may be configured forreceiving the various inputs from a database or other source. Thecritical point model computation module 214 uses the inputs received viathe input receiving module 212 and processes the airfoil using the CPMand provides a calculated velocity flow field as a result with theprescribed numerical resolution. The calculated velocity flow field isconsequently processed by the flow field calculation module 216 toobtain the pressure field and airfoil performance characteristics. Thevelocity/pressure profiles along with the FBPs may be displayed via adisplay 218 with the prescribed spatial resolution.

FIG. 4 depicts an embodiment of a CPM M300 for obtaining the aerodynamiccharacteristics of a wing section with a certain airfoil geometry andspecified α. The CPM M300 begins at S305. At S310, the airfoil geometryis specified along with the velocity and angle of attack α, and at S315,a potential flow solver is established. At S320, the initial estimatefor the LESP is obtained for the prescribed a by assuming a conventionalKutta condition at the trailing edge for the upper and lower surfaces.At S325 a viscous model is applied. For example, boundary layer methodsexist in the art to predict flow separation as a function of thepressure gradient. Using a viscous model, boundary layer calculationsare then computed for the resulting pressure distribution to obtain anew location for the trailing edge FSP on the upper surface. At S330,the LESP is then computed for the new FSP and α. Steps S325 and S330 arerepeated until the LESP and FSP converge to within prescribed limits.When they have converged, the resulting velocity distribution, pressuredistribution, aerodynamic lift/moment/drag coefficients are calculatedat S335 using techniques known in the art. The method ends at S340.

The CPM method will now be described in more detail. We considerpotential flow of incompressible fluid past an airfoil. The incidentvelocity vector is at angle alpha to the x-axis. The x-axis is alignedwith the airfoil chord, with the leading edge taken as the origin of thecoordinate system. We assume that the flow separates on the uppersurface (suction side) of the airfoil (see FIG. 1). As a result of flowseparation, a free streamline forms downstream of the separation point.We shall further assume that the pressure stays constant along the freestreamline being equal to the free-stream pressure. We also assume thatthe flow separates at the trailing edge on the lower (pressure side) ofthe airfoil followed by a constant pressure free streamline. Thepressure is also assumed to remain constant along this streamline at thefree stream value.

The potential flow is described by Laplace equations for the velocitypotential or stream function. The flow can also be represented by asingle complex potential w=φ+iψ, where φ is the potential function and ψis the stream function, and z=x+iy, x-axis being oriented along theairfoil chord, with the LE as the origin of the coordinate system. Withthe flow parameters satisfying the Cauchy-Riemann conditions, thecomplex velocity potential is expressed asƒ(z)=dw/dz=u−ivHere, u and v are the velocity components along the x and y-axesrespectively:u=∂φ/∂xv=∂φ/∂yThe problem is solved in the complex plane using conformal mapping.

The function f(z) is analytic everywhere outside the airfoil except atcertain singular points. The solution method generally revolves aroundhow we treat the singularities at the leading edge stagnation point andthe trailing edge separation. The leading edge (LE) singularity isusually taken care of by assuming the origin for the z-plane at theLESP. The trailing edge (TE) singularity is usually resolved by imposingzero velocity and requiring that the flow leave the TE smoothly. Thiscondition is the well-known Kutta condition.

Theodorsen and Garrick described an exact method to solve this problemfor arbitrary airfoil geometry through a series of transformations.There have been other similar methods based on conformal mapping tosolve this problem. Kutta condition plays a pivotal role in all theseapproaches. While differing in the choice of mapping function and howthey finally introduce viscous effects, they all share one commonassumption: Kutta condition at the trailing edge. Flow separation is notprescribed in their potential flow formulation.

The methods of the present invention differ from these approaches andassume that flow separation occurs on the upper surface (suction side)at all positive incidences. The lower surface is treated exactly as inthe conventional approaches, with the flow smoothly leaving the trailingedge.

The problem is solved by seeking a function f(z) that is analyticeverywhere outside the airfoil surface and the separated flow region,and satisfies the (Dirichlet or Neumann) boundary conditions on theairfoil surface and constant pressure conditions along the separatedstreamlines. In the case of separated flow, there are four criticalpoints (singularities) in the flow: the LESP of the airfoil, the FSP,trailing edge and the wake flow at infinity.

The complex function is evaluated in the neighborhood of eachsingularity. If necessary, an auxiliary function is chosen to deal witha particular singularity while maintaining continuity of the function.The Cauchy integral is evaluated throughout complex plane followingstandard techniques in complex variables. The integral is zero at allpoints except at the singularities where its value is given by theprincipal value of the integral or residues at each singularity.

The CPM is based on an analysis of the unknown complex potentialfunction. The objective of the analysis is to determine all the criticalpoints (zeros and singularities) in the flow region and correspondinglyin the region of change of the parametric variable. There is a widevariety and flexibility of choices for selecting the region of change inparametric variable. The selection is made in such a manner that thebehavior of the parameters in this region is well known. For example,the common types of such regions are circles, semi-circles, plane,half-plane, etc. The choice is generally dictated by the specific natureof the singularities in the problem.

Here we describe Kirchoff's method, which is a well-known example forthe extreme case of flow separation from the leading edge of a flatplate. While the case of an airfoil with flow separation at arbitrarylocation is possible, we give this example mainly to describe themathematical approach, which is common to both. We follow the approachdescribed in Gurevich, M. I.: Theory of Jets in Ideal Fluids, AcademicPress, NY and London (1965) (“Gurevich Reference”).

It is assumed that the velocity V (and pressure, p) is constant alongthe free streamline. The flow downstream of the separation point on theupper and lower surfaces are assumed to be free streamlines. In lieu ofdetermining the complex potential was a function of the complex variablez, it is possible to find the function:ζ(w)=dz/dwIf ζ(w) is found, then with a single integration, it is possible findz(w), since,Z=(1/V)∫V(dz/dw)dw=(1/V)∫ζ(w)dw

It is then possible, in principle, to find w(z) by inverting z(w). Thisinversion process is cumbersome and difficult. Hence we revert to theproblem of finding ζ(w). The geometric significance of this functionlies in the fact that it gives a conformal transformation of the regionof change of w onto the region of change of ζ.

The complex potential is determined only up to an arbitrary constant C,where C is complex (C=Cr+iCi). The real part of the arbitrary constantCr is chosen so that the velocity potential φ=0 at the leading-edgestagnation point (a critical point), where the velocity is zero. Theimaginary part of the arbitrary constant Ci is chosen so that thebifurcated streamline along the airfoil surface is represented by ψ=0.Thus the LESP is the origin for the w-plane. It can be easily shown thatφ varies from −∞ to 0 as the flow comes from infinity to the LESP, andchanges from 0 to ∞ along the streamline to downstream infinity.

For every point in the z-plane there is a corresponding point in thew-plane. Also, for every point in the w-plane there is a correspondingpoint in the z-plane except those points on the real, positivesemi-axis. This occurs because the stream function is zero (φ=0) alongboth the dividing streamlines downstream of the LESP but the potentialfunction (φ) is different along these lines. This problem is usuallyresolved by making a branch cut along the positive semi-axis (ψ=0, φ>0).The upper branch of this cut is used to describe the parameters on theupper surface and the lower branch is used for the lower surface. Indoing so, it is critically important to pay attention to the changes inthe argument θ, of the complex functions. Here, θ is the angle betweenthe velocity vector and the x-axis at a given point on the airfoilsurface.

The behavior of the complex function in the flow region is used todetermine a mapping function that correctly characterizes the criticalregions. The conformal transformation of w onto ζ is found with the helpof several successive mapping functions that satisfy all the boundaryconditions and critical points to finally arrive at an explicitrelationship between ζ and w. Because ζ(w) and therefore z(w) are known,it is not necessary to invert z(w). It is possible to use the directrelation between z and ζ as a function of w. Thus w provides thepotential function (φ) and stream function (ψ) in the entire region andin turn they provide the velocity field. Bernoulli's equation is used toobtain the pressure field. The aerodynamic coefficients are determinedcompletely in terms of the flow parameters like density, free streampressure, and free stream velocity for a given airfoil geometry.

The final result is expressed in the form of an integral equation thatrelates the LESP, FSP and alpha for a given airfoil geometry. Thisequation can be solved numerically to provide a unique functionalrelationship between alpha, LESP and FSP for a given airfoil geometry.

Once the LESP and the flow field (velocity and pressure distribution)are determined, it is possible to use boundary layer approximations toinvestigate the viscous effects. For example, the pressure distributionfrom the airfoil LESP can be used to compute transition region andseparation location using standard techniques and compare with assumedseparation location. The pressure distribution is used to computeaerodynamic coefficients. Aerodynamic coefficients can also be computeddirectly from the velocity potential function using the standard Blasiusformula. See Gurevich Reference.

The methods of this invention differ from prior art in a number ofimportant ways:

-   -   1. Flow separation point is included in the potential flow model    -   2. In the main embodiment of the invention, free streamline        model is assumed for the wake; i.e., constant pressure is        assumed along the separated streamline.    -   3. Airfoil geometry is not changed to account for the changes        introduced by viscous effects. Instead, the flow field is first        computed for the specified critical points (LESP, FSP) or alpha        and resulting pressure distribution is used to compute the        boundary layer characteristics. Three possibilities are        addressed depending on the specified initial values:        -   a. Only alpha is specified: When only alpha is specified, it            is necessary to determine both the LESP and FSP. The Kutta            condition is used as the initial guess for FSP at the TE.            The pair of alpha and FSP is used to determine LESP location            as well as the velocity and pressure distribution. The            pressure distribution is then used to compute the boundary            layer characteristics using standard models for predicting            transition and flow separation locations. If the computed            FSP location is different from the assumed TE location, the            new LESP location is computed using the pair of alpha and            FSP to solve the potential flow solver. The procedure is            repeated till the FSP converges to a point with acceptable            level of accuracy. The aerodynamic coefficients are computed            for the resulting flow field.        -   b. Alpha and LESP are specified: This case is relatively            simple. These two values are inserted into the potential            flow solver to obtain the aerodynamic coefficients.        -   c. LESP and FSP are specified: This case too is relatively            straightforward. The values of the LESP and FSP are inserted            into the potential flow solver to obtain the aerodynamic            coefficients.        -   d. CL and alpha are specified: This case is similar to the            Case a, above. LESP and FSP are obtained in an iterative            manner. This problem arises when there is a need to generate            a desired CL at a specified alpha. The program provides            unique combinations of LESP and FSP for any CL (or alpha).            The desired combinations could be established through            effective flow control actuators or by morphing the airfoil            geometry in an iterative manner.

FIG. 5 depicts a method M400 of designing wing sections using CPM. Themethod is based on standard aerodynamic design optimization methods;however, CPM is used in place of other computational methods to measureairfoil characteristics because of the improved accuracy and reducedcomputational burden. A description of the method M400 is as follows.The method M400 begins at S405. At S410, an initial airfoil surfacegeometry is provided, and at S420, one or more flow parameters (e.g., α,velocity) is/are specified. Airfoils are usually designed for a givenrange of α. At S430, a cost function is specified along with anyconstraints. The cost function is a performance metric that is supposedto be minimized (or maximized depending on the metric). For example,suppose we want to obtain a target pressure distribution, then the costfunction is

$J = {\frac{1}{2}{\int_{S}{\left( {p - p_{d}} \right)^{2}\ {\mathbb{d}s}}}}$where S is over the airfoil surface, p_(d) is the desired pressure, inwhich we desire to minimize J subject to any constraints, Q. Constraintsmay be provided in terms of fixed lift coefficient, airfoil thickness,curvature limits, etc. After prescribing the cost function J &constraints Q, providing the initial airfoil geometry, and flowparameter input such as freestream velocity, the method determines theoptimal airfoil by iterating over variations of airfoils that satisfythe constraints. This process is started at S440 where theaforementioned CPM is used to calculate airfoil characteristics and theresulting flow field and performance characteristics (CL, CD, CM andC_(P) distribution). At S450 a cost is computed using the cost function(J). When the cost function J is greater than a prescribed threshold ε,the method generates at S460 a revised airfoil geometry within thespecified constraints. Steps S440 and S450 are then repeated and a newcost function calculated. The optimal airfoil geometry is reached whenthe cost function J is below the prescribed threshold. The output atS470 is an optimal airfoil geometry provided in terms of line segmentsin non-dimensional x/c & y/c airfoil coordinates, where c is the chordlength. The method ends at S480.

The method M400 may be carried out using a suitable processing systemsimilar to that described in FIG. 3 but with the addition of a costfunction module and an airfoil geometry generation module.

The block diagram in FIG. 6 illustrates a control system according to anembodiment of the invention that is similar to a control systemdescribed in U.S. patent application Ser. No. 12/117,010 ('010application), which is incorporated herein by reference in its entirety.FIG. 4 of the '010 Application depicted a system 100 for controllingresponse to aeroelastic effects on a structure immersed in a fluid flowregime. Similar to the system of the '010 application, the system 500 ofFIG. 6 of the present application includes a sensor array 520, which mayinclude a set of flow sensors 524 and a set of structural sensors 522applied to or incorporated into the structure 510. The flow sensor set524 may include one or more FBP sensors such as hot-film sensors or“direct” shear sensors as described in the '010 application. The flowsensor set 524 may also include pressure sensors. The sensor array 520may also include a set of structural sensors 522, which may includeaccelerometers and/or strain gages. The various flow sensors andstructural sensors are disposed on and within the structure 510 to as toprovide information sufficient to establish the flow and structuralstate of the structure 510.

The system 500 further includes a data processor 530 in communicationwith the sensor array 520. The data processor 530 is configured forreceiving and processing data from the various sensors of the sensorarray 520, for determining one or more actuator command signals, and fortransmitting such actuator command signals to one or more aircraftactuators. The data processor 530 may also be configured to receivecommand inputs from a separate source (e.g., pilot input) for use indetermining actuator command signals. The data processor is,accordingly, in communication with such aircraft actuators, which mayinclude control surface actuators 540, structural actuators 550, andsurface flow control actuators 560. Actuation of one or more of theseactuators 540, 550, 560 results in a change to either or both of theflow state and the structural state of the structure 510.

As in the system of the '010 Application, the data processor 530 may beconfigured to process the sensor data in various ways including, but notlimited to, one or more of the actions of summing, shaping, filtering,switching, gain scheduling, limiting, conditioning, and converting. Thedata processor 530 may be configured to operate as a neural network tofacilitate sensor data and command signal processing.

The control system 500 extends the '010 application system to include anairfoil geometry optimizer 532 configured to carry out an optimizationmethod based on the CPM and a cost function. Thus, the data processor530 can determine, not just the conventional surface flow and structuralactuation, but the actual geometry of wing sections that can be setusing airfoil geometry actuators. Airfoil geometry actuators may includeany number of wing morphing technologies known in the art, including,but not limited to, compliant structural actuation, scissor linkagemechanisms, or adaptable wing sails. The data processor 530 can butgenerally need not calculate the optimal airfoil geometry in real-time,as the airfoil may be pre-calculated and stored into a table. The dataprocessor 530 can choose an optimal airfoil geometry based on themission requirements and the current aeroelastic state.

The first step is to calculate the aerodynamic characteristics of thecurrent iterated airfoil geometry. Using the aerodynamiccharacteristics, the cost function J is evaluated. If the cost functionis within some prespecified threshold epsilon, then the cost function isconsidered minimized. In the case of the target pressure profile, thedifference between the target and actual pressure profile will bereduced to within a threshold value. If the cost function J is above thethreshold, then an airfoil geometry variation different from that of anyprevious iteration is generated. Ideally, the variation is generatedsuch that the next iterate is closer to the global minimum. In any case,a variety of optimization methods may be used to generate the nextvariation including, but not limited to, gradient-based and evolutionaryalgorithms. The loop continues until the cost function is below thethreshold, wherein the output is the optimal airfoil geometry.

FIG. 7 depicts a method M600 of obtaining the aerodynamiccharacteristics of a wing section with a certain airfoil geometry giveneither α & LESP; LESP & FSP; or α & FSP as measurements. Given aprescribed airfoil geometry, two of the three parameters α, LESP or FSPare measured, and, using the potential flow solver through CPM, theaerodynamic characteristics of the airfoil geometry are obtained. Themethod M600 begins at S605 and at S610, the airfoil geometry isprescribed. At S620, a combination of two of the three flow parameters(α, LESP or FSP) are measured or calculated from other flowmeasurements. At S630, a potential flow solution is calculated using theCPM. At S640, the flow solution is used to calculate the aerodynamiccharacteristics of the airfoil geometry. The method ends at S645.

Although aerodynamic characteristics like CL and CM can be calculatedfrom two of the three parameters in analytical/numerical form, forimproved response time, the relationship may be stored in apre-calculated table that is a function of the measured pair ofparameters and airfoil geometries, e.g. LE/TE flap deflections. Otherapproximations including, but not limited to, neural networks, could beimplemented to reduce the computational complexity of calculating theaerodynamic characteristics of the wing section. Even though thedescribed potential flow model is not necessarily valid for unsteadyflows, the instantaneous values of two of the three parameters could beused to calculate, as an initial guess, the instantaneous aerodynamiccharacteristics.

In addition, a state observer (e.g. a Kalman filter) could beimplemented to obtain estimates of other flow parameters based on theestimates of these aforementioned flow parameters. As an example, in anaeroelastic system encountering gusts, α, LESP and FSP may be estimatedthrough a combination of inertial/GPS, air data and flow measurementsthrough the use of a observer to separate the effect of gusts, from theaeroelastic pitch/plunge, freestream angle-of-attack and unsteadyaerodynamic effects. The output from the observer can then be used fordetermining the aeroelastic system state and as an output for acontroller to optimize the aeroelastic system for aerodynamicperformance, safety and ride quality, e.g., gust load alleviation.

Although the previous discussions have been primarily focused ontwo-dimensional flows, the invention can be extended tothree-dimensional flows. As an example, sectional airfoil aerodynamiccharacteristics along the span may be calculated using theinstantaneous, sectional LESP & FSP. Similarly, CPM may be extended toinclude chordwise and spanwise potential flows with defined attachmentand separation lines instead of points. Specifically, the invention isapplicable to three-dimensional flows for missiles and other projectilesin which the aerodynamic characteristics depend on the attachment andseparation lines. In addition, an extension to the invention is thecalculation of stability & control derivatives (derivatives of CL, CM,CD with respect to α, LESP and their respective time derivatives) fromCL, CM, CD determined through CPM.

It will be readily understood by those persons skilled in the art thatthe present invention is susceptible to broad utility and application.Many embodiments and adaptations of the present invention other thanthose herein described, as well as many variations, modifications andequivalent arrangements, will be apparent from or reasonably suggestedby the present invention and foregoing description thereof, withoutdeparting from the substance or scope of the invention.

Accordingly, while the present invention has been described here indetail in relation to its preferred embodiment, it is to be understoodthat this disclosure is only illustrative and exemplary of the presentinvention and is made merely for the purposes of providing a full andenabling disclosure of the invention. Many modifications to theembodiments described above can be made without departing from thespirit and scope of the invention. Accordingly, the foregoing disclosureis not intended to be construed or to limit the present invention orotherwise to exclude any other such embodiments, adaptations,variations, modifications and equivalent arrangements.

1. A computer implemented method of calculating flow performancecharacteristics of a body immersed in a fluid under a set of fluid flowconditions, the method comprising: providing a geometrical descriptionof a surface of the body; determining the set of fluid flow conditionsincluding angle of attack without either leading edge stagnation pointlocation or flow separation point location; and calculating using acomputer, a velocity flow field for the body using a critical pointpotential flow methodology; and calculating using the computer, flowperformance characteristics including at least one of a set consistingof angle of attack, leading edge stagnation point location, flowseparation point location, lift coefficient, drag coefficient andpressure coefficient distribution, wherein the action of calculating avelocity flow field includes: determining an estimated flow separationpoint location; determining an estimated leading edge stagnation pointlocation based on the angle of attack and the estimated flow separationpoint location; calculating an estimated pressure distribution;determining an updated flow separation point location; comparing theupdated flow separation point location to the estimated flow separationpoint location; responsive to a determination that the updated flowseparation point location differs from the estimated flow separationpoint location point location by more than a predetermined level,revising the estimated leading edge stagnation point location based onthe updated flow separation point location; repeating the actions ofcalculating an estimated pressure distribution, determining an updatedflow separation point location and comparing the updated flow separationpoint location to the estimated flow separation point location until theupdated flow separation point location differs from the estimated flowseparation point location by less than the predetermined level.
 2. Asystem for determining flow performance characteristics of a bodyimmersed in a fluid under a set of fluid flow conditions, the systemcomprising: a performance data processor including an input receivingmodule adapted for receiving a body surface geometry description, fluidflow parameters, and resolution input; a critical point modelcomputation module in communication with the input receiving module, thecritical point model computation module being adapted for using acritical point potential flow methodology to calculate a velocity flowfield using the surface geometry description, the fluid flow parametersand the resolution input, and a flow field calculation module incommunication with the critical point model computation module, the flowfield calculation module being adapted for determining a pressure fieldaround the body and flow characteristics for the body based on thevelocity field, surface geometry description and fluid flow parameters,wherein the critical point model computation module is further adaptedto calculate the velocity flow field based on fluid flow parameters thatinclude angle of attack without either leading edge stagnation pointlocation or flow separation point location by carrying out the actionsof: determining an estimated flow separation point location; determiningan estimated leading edge stagnation point location based on the angleof attack and the estimated flow separation point location; calculatingan estimated pressure distribution; determining an updated flowseparation point location; comparing the updated flow separation pointlocation to the estimated flow separation point location; responsive toa determination that the updated flow separation point location differsfrom the estimated flow separation point location point location by morethan a predetermined level, revising the estimated leading edgestagnation point location based on the updated flow separation pointlocation; repeating the actions of calculating an estimated pressuredistribution, determining an updated flow separation point location andcomparing the updated flow separation point location to the estimatedflow separation point location until the updated flow separation pointlocation differs from the estimated flow separation point location byless than the predetermined level.